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Full text of "Treatise On Analysis Vol-Ii"

24        XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

Clearly, if / is lower semicontinuous at XG , then  / is upper semicon-
tinuous at this point. Hence we need consider only lower semicontinuous
functions.

If cp is a mapping of a topological space F into E which is continuous at a
point y0 E F, and if /is lower semicontinuous at the point ;c0 = (p(y0\ then
/Q (p is lower semicontinuous at j0 . In particular, if F is a subspace of E, and
if /is lower semicontinuous at a point j0 e F, then so is the function /| F.

Examples

(12.7.1)    A mapping /: E - R is said to have a relative minimum at a point
XQ e E if there exists a neighborhood V of XQ such that/(x) ^/(;c0) for all
x E V. If so, then/is lower semicontinuous at the point x0 . This is clearly the
case whenever f(x0) =  oo.

For each x e R, put f(x) = 0 if x is irrational, and f(x) = \\q if x is
rational and equal to p/q (where p, q are coprime integers and q > 0). For
each integer n ^ 0, the subspace of rational numbers p/q with q g n is closed
in R and discrete. Hence for each irrational x there exists a neighborhood V
of x such that/(j;) :g l/n for all ye V, and therefore /is continuous at the
point x. On the other hand, /has a relative maximum at each rational point,
and is therefore upper semicontinuous on R.

(12.7.2)    A mapping f: E - K w /ower semicontinuous on E z/ flfld ow/y z/, for
each a e R, ?/ze ^eif /""1(]a, +00]) of points x at which f(x) > a w 0/?e 1/2 E
(or, equivalently, the set/""1^ oo, a]) of points x such that/(jc) ^ a is closed
inE).

For to say that / is lower semicontinuous on E signifies that, for each
a e R, the set/"1Qa, +00]) is a neighborhood of each of its points (3.6.4).

If A is any subset of a set E, the characteristic function of A (usually
denoted by (pA) is the mapping of E into R such that (pA(x) = 1 for all x e A
and cpA(x) = 0 for all x e E  A. So we have (pE = 1, q>0 = 0, and the for-
mulas

<PA + <PB = 9 A u B + <?A n B >        9 A. n CB = <?A - 9 A 93 >
(12.7.3)

I<PA - <?B! = <PA o CB + %-n CA 

A                                A                                         A

where A, B are any two subsets of E, and (AA) is any family of subsets of E.